A319594 Number of solutions to dft(p)^2 + dft(q)^2 = (4n-3), where p and q are even sequences of length 2n-1, p(0)=0, p(k)=+1,-1 when k<>0, q(k) is +1,-1 for all k, and dft(x) denotes the discrete Fourier transform of x.

2, 4, 4, 12, 12, 0, 12, 16, 0, 36, 24, 0, 20, 36, 0, 60

Offset

1,1

Comments

Each solution (p,q) corresponds to a family of symmetric Hadamard matrices of size 8n-4. To construct one member from this family, set A = circulant(p) + I, B = circulant(q), C = B, D = A - 2 I and H = [ [A, B, C, D], [B, D, -A, -C], [C, -A, -D, B], [D, -C, B, -A]]. Then A, B, C and D are symmetric and H is Hadamard and symmetric.

Since p and q are assumed to be even, dft(p) and dft(q) are real-valued.

2 divides a(n) for all n. If (p,q) is a solution, then (p,-q) is also a solution.

4 divides a(n) when n>1. If (p,q) is a solution, then (+/-p,+/-q) are also solutions. When n=1, p is the length-1 sequence, (0).

It is known that a(n)>0 for n=25, 26, 29.

Links

Table of n, a(n) for n=1..16.

Jeffery Kline, List of all pairs (p,q) that are counted by a(n), for 1<=n<=16.

Jeffery Kline, List of pairs (p,q) that establish a(n)>0, for n=25, 26, and 29.

Jeffery Kline, Geometric Search for Hadamard Matrices, Theoret. Comput. Sci. 778 (2019), 33-46.

J. Seberry and N.A. Balonin, The Propus Construction for Symmetric Hadamard Matrices, arXiv:1512.01732 [math.CO], 2015.

J. Seberry and N.A. Balonin, Two infinite families of symmetric Hadamard matrices, Australasian Journal of Combinatorics, 69 (2015), 349-357.

Example

For n=1, the a(1)=2 solutions are ((0),(-)) and ((0),(+)). For n=2, the a(2)=4 solutions are ((0,-,-), (-,+,+)), ((0,-,-), (+,-,-)), ((0,+,+),(-,+,+)) and ((0,+,+), (+,-,-)).

Crossrefs

Cf. A007299, A020985, A185064, A258218, A321851, A322617, A322639.

Sequence in context: A319210 A292303 A000936 * A065449 A130618 A129882

Adjacent sequences: A319591 A319592 A319593 * A319595 A319596 A319597

Keyword

nonn,more

Author

Jeffery Kline, Dec 16 2018

Status

approved